The Galvin-Prikry theorem and set existence axioms
type: Item
instance of: scholarly article
title: The Galvin-Prikry theorem and set existence axioms
zbMATH Open document ID: 0667.03043
DOI: 10.1016/0168-0072(89)90066-3
author: Kazuyuki Tanaka
published in: Annals of Pure and Applied Logic
publication date: 1989-01-01T00:00:00Z
review text: This exceptionally efficient paper is written in terms of so-called Reverse Mathematics (RM), a filigree variant of traditional formalist idea(l)s on classifying formulae by equivalences w.r.t. suitable `base' systems (\(\vdash)\). The critique of RM at the end of \textit{A. Macintyre}'s: ``The strength of weak systems'' [Proc. 11th Wittgenstein Symp. 1986 (P. Weingartner and G. Schurz (eds.)), 43-59 (Hölder- Pichler-Tempski, 1987)] queries the then-widespread assumption of RM that its bold face equivalences are `typical'. More precisely, let F and M be prominent propositions in logical (here, set-theoretic) foundations, resp. (here, combinatorial) mathematics, where both F and M contain (set) parameters, which are restricted to some established category in \(F^-\) and \(M^-\). The broad assumption is that, if \(\vdash (F\leftrightarrow M)\) then also \(\vdash (F^-\leftrightarrow M^-)\). (Otherwise, F and M would hardly be `natural' formalizations of the objects meant in natural mathematical language.) By the result stressed in the author's abstract this fails when M is the theorem in the title, F the principle of monotone inductive definitions, and parameters are restricted to the (logical) so-called light face category \(\Sigma^ 1_ 1\). For a less parochial view than RM this is a case of using a sledge hammer to crack or smash a nut since the general ideals of RM were discredited long before they were formulated; cf. artefacts of formal equivalences by Gödel's incompleteness theorem, to be compared to (impeccably precise, and sometimes ingenious) numerical equivalences in numerology, let alone, (temporary) aberrations about theorems being equivalent to the right axioms (since commutativity is right for proving \(5.7=7.5)\). The sledge hammer above has of course other uses, too, even if not in ordinary combinatorics, where F (and hence (M\(\wedge F)\leftrightarrow M)\) is evident. So only \(F\to M\) is significant, \(F\leftrightarrow M\) being then a ritual (modulo…
Mathematics Subject Classification ID: 03F35
zbMATH DE Number: 4091497
zbMATH Keywords: formal provability in weak universal systems
zbMATH Keywords: Reverse Mathematics
reviewed by: Georg Kreisel
MaRDI profile type: Publication
full work available at URL: 0168-0072(89)90066-3
OpenAlex ID: W2136623171
Wikidata QID: Q114684041
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description: scientific article; zbMATH DE number 4091497